College Math Teaching

March 7, 2012

Limits in a first course in multi-variable calculus

Filed under: analysis, calculus, pedagogy — collegemathteaching @ 1:47 am

Question: what is $lim_{(x,y) \rightarrow (1,1)} \frac{x^2 - y^2}{x - y}$? The “obvious” answer is to rewrite $\frac{x^2 - y^2}{x - y} = x + y$ and then just substitute. The mathematical answer is to note that we do get a limit as we approach the point $(1,1)$ from any point in $\Delta \cap D$ where $\Delta$ is a deleted disk and $D$ is the domain of the function, which of course, does not include the line $y = x$. However, most calculus books just speak of the deleted disk without mentioning the domain; the interesting thing here is that the point $(1,1)$ is NOT an isolated (if inessential) singularity.

This is a case in which being too precise might hinder understanding, but not being precise enough can lead to confusion.